In Search of the Bogus Coins

May 10, 2004

RAY: We have seven stacks of coins, each with 100 coins. Real coins weigh ten grams, and phony coins weigh 11 grams. We're going to weigh the coins on an analytic scale, which works just like your average bathroom scale— but it's accurate to within a tenth of a gram.

Here's the rub. Unlike our many fine previous coin puzzlers, in which you have one stack of coins that's counterfeit, this time you could have several stacks of coins that are counterfeit.

TOM: If one coin in the stack is counterfeit, they're all counterfeit?

RAY: Yes. But, it could be that none, some or all of the stacks of coins are bogus. You don't know.

The question is, what's the fewest number of weighings you need to make, to determine which of the stacks, if any, has counterfeit coins?

TOM: And how come it's only one weighing?

RAY: Okay, wise guy. That's right. That's part two of the puzzler. How come it's only one weighing—and how are you going to do it?

Answer:

RAY: Here's how we're going to do it. We're going to take one coin from the first stack, and two coins from the second stack.

TOM: Then four, then eight.

RAY: There you go.

TOM: Sixteen, thirty-two, and sixty-four coins from the last stack.

RAY: Right. So no matter what weight you wind up, it's unique.
For example, if the pile you're weighing is seven grams over, the only way that could happen is if pile one, pile two, and pile three were overweight. If the pile were 63 grams over, that would mean all the stacks were bogus, except the last one.

Brilliant, eh?

TOM: Very, very good.

RAY: So who's our winner?

TOM: The winner is Tabetha Garrison from Harrisonburg, Virginia. And for having her answer selected at random from both of the right answers that we got, Tabetha's going to get a 26-dollar gift certificate to the Shameless Commerce Division of Cartalk.com. And with that certificate, Tabetha, you can knock off your Fathers' Day shopping and pick up a Car Talk necktie. It looks like a normal striped silk tie, but when you get up close if you can read between Dad's mustard stains you'll actually see that the stripes are made up of the names in small print, names from the Car Talk's euphonious list of credits.